Quadrupole ion traps are widely used to record mass spectra by a variety of methods. The most widely used methods employ the mass selective instability scan in which ions are destabilized and ejected from the trap in order of mass/charge. Normally the amplitude of the trapping radiofrequency waveform is ramped to record the mass spectrum but the frequency can also be ramped. For ion motion in the z-direction and external ion detection, the Mathieu equation, which connects points in dimensionless Mathieu parameter az,-qz space with device size (z0), operating conditions (amplitude V, angular frequency of rf Ω) and ion properties (m/z), the relationship in Eq. 1 applies. At the z-axis stability boundary, qz=0.908, ions become unstable and are ejected, so that an m/z scan is accessible through ramping the rf amplitude (V) or altering the rf angular frequency (Ω).m/z=4Vrf/[Ω2z02qz]  Eq. 1
A variant on the RF amplitude method of scanning ions from a trap termed “resonance ejection” uses a small supplementary ac signal to impose a second working point or “hole” on the q axis, i.e. instability can be caused at any arbitrary m/z (and qz) value. Trapped ions can then be scanned through this operating point and ejected in order of increasing m/z as the rf amplitude Vrf is increased. This method of resonance ejection is well-established. It shows significantly increased mass resolution compared to simply scanning across the instability boundary (qz=0.908) and is also useful as a method of increasing the mass/charge range of the ion trap (the increase is the ratio of 0.908 to the value of q at the new instability point).
Eq. 1 also forms the basis for so-called digital ion traps in which the frequency of the drive rf is scanned. However there is a different type of frequency scan which also gives mass spectra, the secular frequency scan. Ion stability in characteristic dimensions r and z is usually expressed in terms of dimensionless Mathieu parameters a and q. The stability condition can also be expressed in terms of Mathieu β parameters. The relationship between the ion secular frequency and the Mathieu β parameter is:ωz=βzΩ/2  Eq 2
A mass spectrum can be recorded by ramping the frequency of the supplementary ac so that resonance is possible with ions of different secular frequencies. This experiment, known as a secular frequency scan, has the practical advantage of not requiring a scan of the main rf amplitude or frequency, and hence using simpler instrumentation. This allows a relatively low amplitude dipolar frequency to be swept to produce a resonance with ions having different m/z values (and hence different secular frequencies). This experiment can be thought of as scanning a secular frequency “hole” through all possible qz values by ramping the frequency of the supplementary ac.